# Talk:Likelihood-ratio test

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## Added General Audience Introduction and Created Examples Contents

The instructions for creating less technical articles suggest starting with a simplier explanation upfront and then get into the technical details later. With a table of contents, the instructions indicate that provides for having something accessible for those that haven't extensively studied this topic; while at the same time, leaving a meaty article for those interested in something more sophisticated. I dont know if I pulled it off perfectly, but I think it improves the article in a way in which who ever put the "to technical" banner would approve.

At the same time I moved the example into its own contents tab to seperate it from the theory portion.

Jeremiahrounds 18:59, 20 June 2007 (UTC)

Would it be possible for any one to add a proof of why the test follows a chi squared distribution ? —Preceding unsigned comment added by Thedreamshaper (talkcontribs) 20:52, 17 February 2010 (UTC)

I think the introduction might benifit from a re-write, perhaps this formula would be more appropriate than the asymptotic version: $\Lambda (x)={\frac {\sup\{\,L(\theta \mid x):\theta \in \Theta _{0}\,\}}{\sup\{\,L(\theta \mid x):\theta \in \Theta \,\}}}.$ --131.111.243.37 (talk) 10:18, 25 May 2010 (UTC)

I added a non-technical description about when these tests arise in practice to the first paragraph. Not an expert, but using this page without something like that was not helpful. —Preceding unsigned comment added by 98.143.103.218 (talk) 04:07, 29 September 2010 (UTC)

## difficult take on the likelihood viewpoint?

I believe this essentially obscures the idea here:

$\Lambda (x)={\frac {\sup\{\,L(\theta \mid x):\theta \in \Theta _{0}\,\}}{\sup\{\,L(\theta \mid x):\theta \in \Theta \,\}}}.$ The likelihood ratio test is the ratio of the probability of the result GIVEN the maximum likelihood estimator in the domain of the null and alternative hypothesis.

The supremums in that equation sort of combine the maximum likelihood method into the theory of likelihood ratios.

I am not making this up. For example, the text Hoel, Introduction of Statistical Theory uses L(x| theta0) / L(x | theta) where each theta is the maximum likelihood estimate applicable to each hypothesis.

You can more simply state it as Hoel does and just note that the thetas are produced by maximum likelihood estimates. So the supremum doesnt need to appear in the theory of likelihood ratios. Then you get a ratio of probabilities that is easier to read and even think about.

I actually initially called the offered equation an error. But that is a bridge to far I think. Putting the supremums in the context where you appear to be maximizing something after the data is taken isnt very useful for understanding the actual method though.

Jeremiahrounds 12:11, 20 June 2007 (UTC)

I don't think there is any maximum involved in the Likelihood-ratio test, you just have to make the ratio of the likelihood under hypothesis H0 and H1. I'm not an expert in statistics but I think this equation introduces a confusion between Likelihood-ratio test and maximum likelihood estimation. I have never seen it presented this way anyway... Sylenius 14:45, 27 June 2007 (UTC)

I think Jeremiahrounds is mistaken. In case the MLEs actually exist, the likelihood-ratio test statistic is in fact equal to what Hoel's book says it is, and also it is equal to the expression in TeX above, which appears in this article. But the likelihood-ratio test statistic can exist even in cases where MLEs don't exist, simply because the sup exists and the max does not, i.e. the sup is not actually attained. Moreover, the problem of non-unique MLEs doesn't matter, since it is only the value of the sup rather than the value of θ where the sup occurs that matters. Michael Hardy 19:05, 27 June 2007 (UTC)

## Untitled

I may get to that if someone doesn't beat me to it. Hundreds of articles here are in need of TeX to replace what was used here before 2003. Michael Hardy 22:57 Feb 2, 2003 (UTC)

The article uses λ in some places, and Λ in others -- is this intentional, or should they all be one or the other?

(Capital) Λ is the most frequently used notation for the test statistic. Michael Hardy 20:12 Feb 4, 2003 (UTC)

Can the Likelihoor ratio test be used in place of the F-test for a fixed effects models. Any diffrences from the F-test in this case? What about using LRT for testing fixed effects in mixed model?

The F-test is the likelihood ratio test in such models. Michael Hardy 22:30, 3 September 2005 (UTC)

Hi. I may be misguided or mistaken here, I'm hardly expert. But I think the definition of the test statistic given is inconsistent with the test statistic given. The unrestricted numerator will be larger than the restricted denominator, so the ratio will be greater than 1, and its log will be positive, so -2 log Λ will be negative and can hardly be chi-square distributed. I think that either the ratio should be inverted, or the test statistic multiplied by negative 1, to keep things consistent. (My apologies again if I'm making a basic mistake, a possibility of which the likelihood is high.) Stevewaldman (talk) 00:58, 20 January 2008 (UTC)

## "asymptotically"

"If the null hypothesis is true, then −2 log Λ will be asymptotically χ2 distributed" The validity conditions of this theorem should be given. "asymptotically" when what tends to what value ?

I have now answered this question in the article. Michael Hardy 02:36, 28 October 2005 (UTC)
There's really no further restriction on the random variables ("n independent identically distributed random variables")? Dchudz 15:22, 13 July 2007 (UTC)